Some local definability theory for holomorphic functions

Abstract

Let \mathcal{F} be a collection of holomorphic functions and let \mathbb{R}(PR(\mathcal{F})) denote the reduct of the structure \mathbb{R}_{an} to the ordered field operations together with the set of proper restrictions (see below) of the real and imaginary parts of all functions in \mathcal{F}. We ask the question: Which holomorphic functions are locally definable (ie have their real and imaginary parts locally definable) in the structure \mathbb{R}(PR(\mathcal{F}))? It is easy to see that the collection of all such functions is closed under composition, partial differentiation, implicit definability (via the Implicit Function Theorem in one dependent variable) and Schwarz Reflection. We conjecture that this exhausts the possibilities and we prove as much in the neighbourhood of generic points. More precisely, we show that these four operations determine the natural pregeometry associated with \mathbb{R}(PR(\mathcal{F}))-definable, holomorphic functions.